Banach February 2009

banach@mathdept.okstate.edu

1 participants
20 discussions

Abstract of a paper by Paul F. X. Mueller
by alspach＠fourier.math.okstate.edu 11 Feb '09

11 Feb '09

This is an announcement for the paper "Extrapolation of vector valued rearrangment operators II" by Paul F. X. Mueller. Abstract: We determine the extrapolation law for rearrangement operators of the Haar system on vector valued Hardy spaces. Archive classification: math.FA Mathematics Subject Classification: 46B42, 46B70, 47B37 The source file(s), pfxm_2009.bbl: 3113 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0902.1330 or http://arXiv.org/abs/0902.1330 or by email in unzipped form by transmitting an empty message with subject line uget 0902.1330 or in gzipped form by using subject line get 0902.1330 to: math(a)arXiv.org.

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Abstract of a paper by Gelu Popescu
by alspach＠fourier.math.okstate.edu 11 Feb '09

11 Feb '09

This is an announcement for the paper "Noncommutative hyperbolic geometry on the unit ball of $B(H)^n$" by Gelu Popescu. Abstract: In this paper we introduce a hyperbolic distance $\delta$ on the noncommutative open ball $[B(H)^n]_1$, where $B(H)$ is the algebra of all bounded linear operators on a Hilbert space $H$, which is a noncommutative extension of the Poincare-Bergman metric on the open unit ball of $C^n$. We prove that $\delta$ is invariant under the action of the group $Aut([B(H)^n]_1)$ of all free holomorphic automorphisms of $[B(\cH)^n]_1$, and show that the $\delta$-topology and the usual operator norm topology coincide on $[B(H)^n]_1$. Moreover, we prove that $[B(H)^n]_1$ is a complete metric space with respect to the hyperbolic metric and obtained an explicit formula for $\delta$ in terms of the reconstruction operator. A Schwarz-Pick lemma for bounded free holomorphic functions on $[B(H)^n]_1$, with respect to the hyperbolic metric, is also obtained. Archive classification: math.FA math.OA Mathematics Subject Classification: 46L52; 32F45; 47L25; 32Q45 Remarks: 29 pages, to appear in J. Funct. Anal The source file(s), hyperbolic.tex: 116240 bytes, is(are) stored in gzipped form as 0810.0644.gz with size 30kb. The corresponding postcript file has gzipped size 78kb. Submitted from: gelu.popescu(a)utsa.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0810.0644 or http://arXiv.org/abs/0810.0644 or by email in unzipped form by transmitting an empty message with subject line uget 0810.0644 or in gzipped form by using subject line get 0810.0644 to: math(a)arXiv.org.

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Abstract of a paper by Fernando Rambla-Barreno and Jarno Talponen
by alspach＠fourier.math.okstate.edu 11 Feb '09

11 Feb '09

This is an announcement for the paper "Uniformly convex-transitive function spaces" by Fernando Rambla-Barreno and Jarno Talponen. Abstract: We introduce a property of Banach spaces called uniform convex-transitivity, which falls between almost transitivity and convex-transitivity. We will provide examples of uniformly convex-transitive spaces. This property behaves nicely in connection with some Banach-valued function spaces. As a consequence, we obtain new examples of convex-transitive Banach spaces. Archive classification: math.FA Mathematics Subject Classification: 46B04; 46B20; 46B25 The source file(s), RotCad_acc.tex: 46198 bytes, is(are) stored in gzipped form as 0902.0640.gz with size 14kb. The corresponding postcript file has gzipped size 99kb. Submitted from: talponen(a)cc.helsinki.fi The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0902.0640 or http://arXiv.org/abs/0902.0640 or by email in unzipped form by transmitting an empty message with subject line uget 0902.0640 or in gzipped form by using subject line get 0902.0640 to: math(a)arXiv.org.

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Abstract of a paper by Daniel Pellegrino and Eduardo V. Teixeira
by alspach＠fourier.math.okstate.edu 11 Feb '09

11 Feb '09

This is an announcement for the paper "Norm optimization problem for linear operators in classical Banach spaces" by Daniel Pellegrino and Eduardo V. Teixeira. Abstract: We prove a linear operator T acting between l_p-type spaces attains its norm if, and only if, there exists a not weakly null maximizing sequence for T. For 1<p=q we show that any not weakly null maximizing sequence for a norm attaining operator T from l_p to l_q has a norm-convergent subsequence. We also prove that for any fixed x_0 in l_p, the set of operators T from l_p to l_q that attain their norm at x_0 is lineable. The same result is proven for the set of all operators that do not attain their norms. Archive classification: math.FA Mathematics Subject Classification: 46B20 Remarks: 12 pages The source file(s), pell-teix-JFA02Fev09.tex: 35990 bytes, is(are) stored in gzipped form as 0902.0454.gz with size 10kb. The corresponding postcript file has gzipped size 91kb. Submitted from: dmpellegrino(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0902.0454 or http://arXiv.org/abs/0902.0454 or by email in unzipped form by transmitting an empty message with subject line uget 0902.0454 or in gzipped form by using subject line get 0902.0454 to: math(a)arXiv.org.

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Abstract of a paper by Tim Netzer
by alspach＠fourier.math.okstate.edu 11 Feb '09

11 Feb '09

This is an announcement for the paper "Representation and approximation of positivity preservers" by Tim Netzer. Abstract: We consider a closed set S in R^n and a linear operator \Phi on the polynomial algebra R[X_1,...,X_n] that preserves nonnegative polynomials, in the following sense: if f\geq 0 on S, then \Phi(f)\geq 0 on S as well. We show that each such operator is given by integration with respect to a measure taking nonnegative functions as its values. This can be seen as a generalization of Haviland's Theorem, which concerns linear functionals on polynomial algebras. For compact sets S we use the result to show that any nonnegativity preserving operator is a pointwise limit of very simple nonnegativity preservers with finite dimensional range. Archive classification: math.FA math.RA Mathematics Subject Classification: 12E05; 15A04; 47B38; 44A60; 31B10; 41A36 Remarks: 17 pages The source file(s), positivitypreservers.tex: 49618 bytes, is(are) stored in gzipped form as 0902.0279.gz with size 15kb. The corresponding postcript file has gzipped size 99kb. Submitted from: tim.netzer(a)gmx.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0902.0279 or http://arXiv.org/abs/0902.0279 or by email in unzipped form by transmitting an empty message with subject line uget 0902.0279 or in gzipped form by using subject line get 0902.0279 to: math(a)arXiv.org.

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Abstract of a paper by Su Gao, Steve Jackson, and Bunyamin Sari
by alspach＠fourier.math.okstate.edu 03 Feb '09

03 Feb '09

This is an announcement for the paper "On the complexity of the uniform homeomorphism relation between separable Banach spaces" by Su Gao, Steve Jackson, and Bunyamin Sari. Abstract: We consider the problem of determining the complexity of the uniform homeomorphism relation between separable Banach spaces in the Borel reducibility hierarchy of analytic equivalence relations. We prove that the complete $K_{\sigma}$ equivalence relation is Borel reducible to the uniform homeomorphism relation, and we also determine the possible complexities of the relation when restricted to some small classes of Banach spaces. Moreover, we determine the exact complexity of the local equivalence relation between Banach spaces, namely that it is bireducible with $K_{\sigma}$. Finally, we construct a class of mutually uniformly homeomorphic Banach spaces such that the equality relation of countable sets of real numbers is Borel reducible to the isomorphism relation on the class. Archive classification: math.FA math.LO The source file(s), gjs_24.tex: 107663 bytes, is(are) stored in gzipped form as 0901.4092.gz with size 33kb. The corresponding postcript file has gzipped size 163kb. Submitted from: bunyamin(a)unt.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0901.4092 or http://arXiv.org/abs/0901.4092 or by email in unzipped form by transmitting an empty message with subject line uget 0901.4092 or in gzipped form by using subject line get 0901.4092 to: math(a)arXiv.org.

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Abstract of a paper by Yun-Su Kim
by alspach＠fourier.math.okstate.edu 03 Feb '09

03 Feb '09

This is an announcement for the paper "An answer to the invariant subspace problem" by Yun-Su Kim. Abstract: To answer to the invariant subspace problem, we show that every transcendental operator has a non-trivial invariant subspace. Archive classification: math.FA Mathematics Subject Classification: 47A15; 47S99. The source file(s), invariant1.tex: 19822 bytes, is(are) stored in gzipped form as 0901.3852.gz with size 6kb. The corresponding postcript file has gzipped size 58kb. Submitted from: Yun-Su.Kim(a)utoledo.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0901.3852 or http://arXiv.org/abs/0901.3852 or by email in unzipped form by transmitting an empty message with subject line uget 0901.3852 or in gzipped form by using subject line get 0901.3852 to: math(a)arXiv.org.

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Abstract of a paper by Petr Hajek and Antonin Prochazka
by alspach＠fourier.math.okstate.edu 03 Feb '09

03 Feb '09

This is an announcement for the paper "$C^k$-smooth approximations of LUR norms" by Petr Hajek and Antonin Prochazka. Abstract: Let $X$ be a WCG Banach space admitting a $C^k$-Fr\' echet smooth norm. Then $X$ admits an equivalent norm which is simultaneously $C^1$-Fr\' echet smooth, LUR, and a uniform limit of $C^k$-Fr\' echet smooth norms. If $X=C([0,\alpha])$, where $\alpha$ is an ordinal, then the same conclusion holds true with $k=\infty$. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46B03, 46E15 The source file(s), LUR3-13-1-2.tex: 67805 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0901.3623 or http://arXiv.org/abs/0901.3623 or by email in unzipped form by transmitting an empty message with subject line uget 0901.3623 or in gzipped form by using subject line get 0901.3623 to: math(a)arXiv.org.

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Abstract of a paper by Hanfeng Li and Anthony Weston
by alspach＠fourier.math.okstate.edu 03 Feb '09

03 Feb '09

This is an announcement for the paper "Strict p-negative type of a semi-metric space" by Hanfeng Li and Anthony Weston. Abstract: Doust and Weston introduced a new method called "enhanced negative type" for calculating a non trivial lower bound p(T) on the supremal strict p-negative type of any given finite metric tree (T,d). (In the context of finite metric trees any such lower bound p(T) > 1 is deemed to be non trivial.) In this paper we refine the technique of enhanced negative type and show how it may be applied more generally to any finite semi-metric space (X,d) that is known to have strict p-negative type for some non negative p. This allows us to significantly improve the lower bounds on the supremal strict p-negative type of finite metric trees that were given by Doust and Weston and, moreover, leads in to one of our main results: The supremal p-negative type of a finite semi-metric space cannot be strict. By way of application we are then able to exhibit large classes of finite metric spaces (such as finite isometric subspaces of Hadamard manifolds) that must have strict p-negative type for some p > 1. We also show that if a semi-metric space (finite or otherwise) has p-negative type for some p > 0, then it must have strict q-negative type for all q in [0,p). This generalizes a well known theorem of Schoenberg and leads to further applications. Archive classification: math.FA math.MG Mathematics Subject Classification: 46B20 Remarks: 12 pages The source file(s), HLAW-Final.tex: 44858 bytes, is(are) stored in gzipped form as 0901.0695.gz with size 13kb. The corresponding postcript file has gzipped size 353kb. Submitted from: westona(a)canisius.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0901.0695 or http://arXiv.org/abs/0901.0695 or by email in unzipped form by transmitting an empty message with subject line uget 0901.0695 or in gzipped form by using subject line get 0901.0695 to: math(a)arXiv.org.

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Abstract of a paper by Teodor Banica, Benoit Collins, and Jean-Marc Schlenker
by alspach＠fourier.math.okstate.edu 03 Feb '09

03 Feb '09

This is an announcement for the paper "On orthogonal matrices maximizing the 1-norm" by Teodor Banica, Benoit Collins, and Jean-Marc Schlenker. Abstract: For $U\in O(N)$ we have $||U||_1\leq N\sqrt{N}$, with equality if and only if $U=H/\sqrt{N}$, with $H$ Hadamard matrix. Motivated by this remark, we discuss in this paper the algebraic and analytic aspects of the computation of the maximum of the 1-norm on $O(N)$. The main problem is to compute the $k$-th moment of the 1-norm, with $k\to\infty$, and we present a number of general comments in this direction. Archive classification: math.OA math.CO Remarks: 17 pages The source file(s), omx.tex: 34742 bytes, is(are) stored in gzipped form as 0901.2923.gz with size 11kb. The corresponding postcript file has gzipped size 98kb. Submitted from: banica(a)picard.ups-tlse.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0901.2923 or http://arXiv.org/abs/0901.2923 or by email in unzipped form by transmitting an empty message with subject line uget 0901.2923 or in gzipped form by using subject line get 0901.2923 to: math(a)arXiv.org.

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Banach February 2009